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Rayleigh principle for linear Hamiltonian systems without controllability∗

Werner Kratz, Roman Šimon Hilscher (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools...

Real zeros of holomorphic Hecke cusp forms

Amit Ghosh, Peter Sarnak (2012)

Journal of the European Mathematical Society

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity.

Realization theory methods for the stability investigation of nonlinear infinite-dimensional input-output systems

Volker Reitmann (2011)

Mathematica Bohemica

Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the stability and instability investigation of a class of nonlinear Volterra integral equations in a Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in a weighted Sobolev space, which has the same stability properties as the Volterra equation.

Reconstruction of map projection, its inverse and re-projection

Tomáš Bayer, Milada Kočandrlová (2018)

Applications of Mathematics

This paper focuses on the automatic recognition of map projection, its inverse and re-projection. Our analysis leads to the unconstrained optimization solved by the hybrid BFGS nonlinear least squares technique. The objective function is represented by the squared sum of the residuals. For the map re-projection the partial differential equations of the inverse transformation are derived. They can be applied to any map projection. Illustrative examples of the stereographic and globular Nicolosi projections...

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